Solve Nested Radicals: Cube Root of Square Root of 36

Q: Why do we multiply the exponents instead of adding them?

When you have nested operations like a root of a root, you're applying the power rule: $ (x^a)^b = x^{ab} $. The square root gives us $ 36^{\frac{1}{2}} $, then the cube root raises this to the $ \frac{1}{3} $ power.

Q: Can I solve this by calculating the square root first?

Yes! You can work from inside out: $ \sqrt{36} = 6 $, then $ \sqrt[3]{6} = 6^{\frac{1}{3}} $. However, this doesn't match any of the given answer choices, which are all in exponential form with base 36.

Q: How do I remember the formula for nested radicals?

Think of it as "root of root = root times root" in the denominator. So $ \sqrt[m]{\sqrt[n]{x}} = x^{\frac{1}{m \times n}} $. The denominators multiply: 2 × 3 = 6.

Q: What if the numbers were different, like cube root of fourth root?

The same rule applies! $ \sqrt[3]{\sqrt[4]{x}} = x^{\frac{1}{3 \times 4}} = x^{\frac{1}{12}} $. Always multiply the root indices in the denominator of the final exponent.

Q: Why is this answer different from just 36 to some power?

The nested radicals create a compound operation. Each root operation divides the exponent, so two root operations give us $ \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} $, making the final result much smaller than $ 36^{\frac{1}{2}} $ or $ 36^{\frac{1}{3}} $.

Exponent Rules with Nested Radicals

Complete the following exercise:

$\sqrt[3]{\sqrt{36}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:03 A 'regular' root is of the order 2

00:09 When we have a number (A) to the power of (B) in a root of order (C)

00:14 The result equals the number (A) to the root of order (B times C)

00:18 We will apply this formula to our exercise

00:24 Calculate the order multiplication

00:31 When we have a number (A) to the power of (B) in a root of order (C)

00:36 The result equals the number (A) to the power of their quotient (B divided by C)

00:39 We will apply this formula to our exercise

00:43 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt[3]{\sqrt{36}}=$

Step-by-step solution

To solve this problem, let's analyze and simplify the given expression $\sqrt[3]{\sqrt{36}}$ .

Step 1: Identify the root operations. We have a square root, $\sqrt{36}$ , and a cube root, $\sqrt[3]{\ldots}$ .
Step 2: Use the formula for roots for a root of a root: $\sqrt[m]{\sqrt[n]{x}} = x^{\frac{1}{mn}}$ .
Step 3: Apply this formula to the problem. In this case, the first operation is a square root, which can be written as $36^{\frac{1}{2}}$ , and the second operation is a cube root. Therefore, $\sqrt[3]{\sqrt{36}} = (36^{\frac{1}{2}})^{\frac{1}{3}}$ .
Step 4: Simplify using the power of a power rule, which allows us to multiply exponents: $(36^{\frac{1}{2}})^{\frac{1}{3}} = 36^{\frac{1}{2 \times 3}} = 36^{\frac{1}{6}}$ .

Thus, the expression $\sqrt[3]{\sqrt{36}}$ simplifies to $36^{\frac{1}{6}}$ .

Final Answer

$36^{\frac{1}{6}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Root of root equals base raised to product of denominators
Technique: Convert $\sqrt[3]{\sqrt{36}}$ to $36^{\frac{1}{2} \times \frac{1}{3}} = 36^{\frac{1}{6}}$
Check: Power rule works backwards: $(36^{\frac{1}{6}})^6 = 36$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't convert $\sqrt[3]{\sqrt{36}}$ to $36^{\frac{1}{2} + \frac{1}{3}} = 36^{\frac{5}{6}}$ ! This gives a completely different answer because you're combining operations incorrectly. Always multiply the fractional exponents when dealing with nested radicals.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[10]{\sqrt[10]{1}}=$

FAQ

Everything you need to know about this question

Why do we multiply the exponents instead of adding them?

When you have nested operations like a root of a root, you're applying the power rule: $(x^a)^b = x^{ab}$ . The square root gives us $36^{\frac{1}{2}}$ , then the cube root raises this to the $\frac{1}{3}$ power.

Can I solve this by calculating the square root first?

Yes! You can work from inside out: $\sqrt{36} = 6$ , then $\sqrt[3]{6} = 6^{\frac{1}{3}}$ . However, this doesn't match any of the given answer choices, which are all in exponential form with base 36.

How do I remember the formula for nested radicals?

Think of it as "root of root = root times root" in the denominator. So $\sqrt[m]{\sqrt[n]{x}} = x^{\frac{1}{m \times n}}$ . The denominators multiply: 2 × 3 = 6.

What if the numbers were different, like cube root of fourth root?

The same rule applies! $\sqrt[3]{\sqrt[4]{x}} = x^{\frac{1}{3 \times 4}} = x^{\frac{1}{12}}$ . Always multiply the root indices in the denominator of the final exponent.

Why is this answer different from just 36 to some power?

The nested radicals create a compound operation. Each root operation divides the exponent, so two root operations give us $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ , making the final result much smaller than $36^{\frac{1}{2}}$ or $36^{\frac{1}{3}}$ .

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